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comment Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
Finally, I completed all details.
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revised Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
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revised Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
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revised Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
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1d
revised Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
fixing typos x -> z
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answered Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
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comment A conjectured identity for tetralogarithms $\operatorname{Li}_4$
@user153012 Well-order rationals in the open interval $(0,1)$ using lexicographic order $\mathcal L$ on $\langle denominator, numerator\rangle$ pairs. Assign to each identity the maximal (according to $\mathcal L$) argument among its tetralogarithm terms. Select identities that are assigned the least (according to $\mathcal L$) rational number, and among them select identities with the least number of tetralogarithm terms. In case of a tie, select identities whose second largest rational argument is the least, then third largest, etc. The remaining identity is the simplest.
2d
comment Log Log Integrals II
The last formula for $\zeta''\left(\tfrac{1}{2}\right)$ does not check numerically. Are you sure it's correct? Do you have a reference or a proof for it?
2d
comment Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$
I can confirm your results for both integrals. They can be evaluated in Mathematica using an integral representation $\operatorname{Li}_\nu(z)=\frac{z}{\Gamma(\nu)}{\large\int}_0^\infty \frac{t^{\nu-1}}{e^t-z}\!\;dt$ and changing the order of integration.
Jul
28
comment Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$
$$\gamma^{(0,1)} \!\left(1, \tfrac{3}{4} \right) - \gamma^{(0,1)} \!\left(1, \tfrac{1}{4} \right) = \zeta^{(1,0)} \!\left(2, \tfrac{3}{4} \right)- \zeta^{(1,0)} \!\left(2, \tfrac{1}{4} \right)-16G$$
Jul
28
comment A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$
It can be represented as a difference of 3-factor integrals (with powers): ${\Large\int}_0^1\ln(x)\,\ln(2+x)\,\ln^2(1+x)\,dx - {\Large\int}_0^1\ln^2(x)\,\ln(2+x)\,\ln(1+x)\,dx$, that makes it more similar to other integrals you mentioned.
Jul
27
reviewed Leave Open Free and bound variables in quantificational logic
Jul
27
reviewed Close Element of one subspace not part of the other?
Jul
27
reviewed Close Maximum of uniform random variable
Jul
27
reviewed Close Find distance between a plane and some points
Jul
27
asked Are logarithms of prime numbers algebraically independent?
Jul
26
comment Every non-increasing sequence of polynomial towers stabilizes — Finitary proof
Thanks, Andres! I think by a finitary proof you mean a proof in $\mathsf{PA}$. As I mentioned in my question, I do not restrict finitary proofs to $\mathsf{PA}$ only. We could use a set theory restricted to hereditary finite sets, iterated consistensy and soundness schemas, or some other "obvious" finitary axioms (I do not define what "obvious" exactly means, it's subjective of course, but I'd like to see something more simple than the proposition being proved itself).